The Nano chargers are nice and trusty small devices, and because of their low amperes output, quite safe to use also (but they don't stop at 4.20 Volts, so use a voltmeter to securely check and stop the charge of the cells if you want the cells to be fully topped off!).
But they are cheap, and not very high quality!
I have had (and still have) a few of these, and some have had this issue (the LED not coming ON when connected to the grid).
The problem is the metal springs inside that connects to the foldable legs that goes into the power-outlet.
You need to open the charger, carefully pry the plastic-cover open, and bend the metal-springs a bit such as to make them push harder to the foldable legs when you fold the legs out.
I don't recommend this if you have no skills or knowledge with some DIY!
Get someone that does to help you!
But once this procedure is done, they last for ages.